Qubit state copying

ABSTRACT

Systems and methods for copying the classical state of a source qubit to a target qubit are provided. These techniques may be used to read out the states of an array of qubits.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims benefit, under 35 U.S.C. § 119(e), of U.S.Provisional Patent Application No. 60/675,139, filed on Apr. 26, 2005,which is hereby incorporated by reference herein in its entirety.

1. FIELD OF THE INVENTION

The present methods and systems are related to the field of quantumcomputing and, in particular, to superconducting devices.

2. BACKGROUND

In 1982, Richard Feynman proposed that a controllable quantum systemcould be used to simulate other quantum systems more efficiently thanconventional computers. See Feynman, 1982, International Journal ofTheoretical Physics 21, pp. 467-488. This controllable quantum system isnow commonly referred to as a quantum computer, and effort has been putinto developing a general purpose quantum computer that can be used tosimulate quantum systems or run specialized quantum algorithms. Inparticular, solving a model for the behavior of a quantum systemcommonly involves solving a differential equation related to theHamiltonian of the quantum system. David Deutsch observed that a quantumsystem could be used to yield a time savings, later shown to be anexponential time savings, in certain computations. If one had a problem,modeled in the form of an equation that represented the Hamiltonian ofthe quantum system, the behavior of the system could provide informationregarding the solutions to the equation. See Deutsch, 1985, Proceedingsof the Royal Society of London A 400, pp. 97-117.

One limitation in the quantum computing art is the identification ofsystems that can support quantum computation. As detailed in thefollowing sections, a qubit, which is analogous to a “bit” of aclassical digital computer, serves as the basis for storing quantuminformation. However, qubits must be able to retain coherent quantumbehavior long enough to perform quantum computations. The loss ofcoherent quantum behavior is referred to as decoherence. Further,techniques for reading the state of qubits are needed in order todetermine the result of a quantum computation. Ideally, such readoutmechanisms do not introduce decoherence to the quantum computing systemprior to a readout operation.

The computing power of a quantum computer increases as its basicbuilding blocks, qubits, are coupled together in such a way that thequantum state of one qubit affects the quantum state of each of thequbits to which it is coupled. This form of coupling includes the effectreferred to as entanglement. Another limitation in the quantum computingart is the identification of methods that can be used to controllablyentangle the states of qubits without introducing a significant sourceof decoherence.

2.1 Approaches to Quantum Computing

There are several general approaches to the design and operation of aquantum computer. One approach referred to as “circuit model quantumcomputing” is based on a model in which logical gates are applied toqubits, much like bits, and can be programmed to perform calculationsusing quantum logic. This model of quantum computing requires qubitswith long coherence times. Efforts have made to develop robust qubitsthat can perform quantum logic functions. For example, see Shor, 2001,arXiv.org:quant-ph/0005003. However, reducing qubit decoherence inquantum systems to the point that many calculations are performed beforequantum information stored in the quantum system is destroyed has notbeen satisfactorily achieved in the art.

Another approach to quantum computing known as “thermally-assistedadiabatic quantum computing,” involves finding the lowest energyconfiguration of an array of qubits. This approach does not makecritical use of quantum gates and circuits. Instead, it uses classicaleffects, and quantum effects in some cases, to manipulate the states ofa system of interacting qubits starting from a known initial Hamiltonianso that the final state represents the Hamiltonian of the physicalsystem in question. In this process, quantum coherence is not a strictrequirement for the qubits. An example of this type of approach isadiabatic quantum computing. See, for example, Farhi et al., 2001,Science 292, pp. 472-476.

2.2 Qubits

A quantum bit, or qubit, is the quantum mechanical analog of theconventional digital bit. Instead of only encoding one of two discretestates, like “0” and “1” in a bit, a qubit can also be placed in asuperposition of 0 and 1. That is, the qubit can exist in both the “0”and “1” state at the same time, and can thus perform a quantumcomputation on both states simultaneously. Thus, a qubit holding a purediscrete state (0 or 1) is said to be in a classical state, whereas aqubit holding a superposition of states is said to be in a quantumstate. In general, N qubits can be in a superposition of 2^(N) states.Quantum algorithms make use of the superposition property to speed upcertain computations.

In standard notation, the basis states of a qubit are referred to as the|0

and |1

states. During quantum computation, the state of a qubit, in general, isa superposition of basis states so that the qubit has a nonzeroprobability of occupying the |0

basis state and a simultaneous nonzero probability of occupying the |1

basis state. Mathematically, a superposition of basis states means thatthe overall state of the qubit, denoted |Ψ

, has the form |Ψ

=a|0

+b|1

, where a and b are coefficients corresponding to the probabilities |a|²and |b|² of obtaining a |0

or |1

upon measurement, respectively. Coefficients a and b each have real andimaginary components. The quantum nature of a qubit is largely derivedfrom its ability to form a coherent superposition of basis states. Aqubit is in a coherent superposition as long as the amplitudes andphases of coefficients a and b are not affected by the outsideenvironment. A qubit will retain this ability to exist as a coherentsuperposition of basis states when the qubit is sufficiently isolatedfrom sources of decoherence.

To complete a computation using a qubit, the state of the qubit ismeasured (e.g., read out). Typically, when a measurement of the qubit isdone, the quantum nature of the qubit is temporarily lost and thesuperposition of basis states collapses to either the |0

basis state or the |1

basis state, thus regaining its similarity to a conventional bit. Theactual state of the qubit after it has collapsed depends on theprobabilities |a|² and |b|² immediately prior to the readout operation.For more information on qubits, generally, see Nielsen and Chuang, 2000,Quantum Computation and Quantum Information, Cambridge University Press,Cambridge, pp. 344-345.

2.3 Superconducting Qubits

There are many different technologies that can be used to build quantumcomputers. One implementation uses superconducting materials.Superconducting qubits have the advantage of scalability. Thepossibility of realizing large scale quantum computers usingsuperconducting qubits is promising since the technologies and processesinvolved in fabricating superconducting qubits are similar to those usedfor conventional silicon-based computers, for which there already existsinfrastructure and technological know-how. Toward the realization ofsuch a computer, Shnirman et al., 1997, Physical Review Letters 79,2371-2374, proposed a superconducting quantum computer using Josephsonjunctions to produce the required quantum effects.

Superconducting qubits can be separated into several categoriesdepending on the physical property used to encode information. A generaldivision of qubits separates them into charge and phase devices, asdiscussed in Makhlin et al., 2001, Reviews of Modern Physics 73, pp.357-400.

A superconducting qubit is typically characterized by two differenttypes of energy, charging energy E_(c), and Josephson energy E_(J). Themagnitude of each of these energy types in a given superconducting qubitdepends on the physical parameters of the qubit. For example, thecharging energy of a superconducting qubit is a function of the chargingenergies of the components (e.g., qubit junctions) of the qubit. Thecharging energy of a qubit junction, in turn, is defined as e²/(2C),where C is the capacitance of the junction. The Josephson energy of asuperconducting qubit is a function of the Josephson energies of thecomponents (e.g., qubit junctions) in the qubit. The Josephson energy ofa qubit junction (e.g., Josephson junction), in turn, is related to thecritical current of the qubit junction. Specifically, the Josephsonenergy of a qubit junction is proportional to the critical current Ic ofthe junction and satisfies the relationship E_(J)=(

/2e)I_(c), where

h is Planck's constant divided by 2π. The ratio of the overall Josephsonenergy and the overall charging energy of a superconducting qubit can beused to classify superconducting qubits. For example, in oneclassification scheme, when the overall charging energy of a givensuperconducting qubit is much greater than the overall Josephson energyof the qubit, the qubit is deemed to be a charge qubit. And, when theoverall Josephson energy of a given superconducting qubit is muchgreater than the overall charging energy of the qubit, the qubit isdeemed to be a phase qubit. As used herein, the term “much greater” inthe context of evaluating two energy terms means that one energy termmay be anywhere from two times greater to more than twenty times greaterthan the second energy term.

In quantum systems based on qubits, phase and charge are conjugatevariables. That is, a higher accuracy of determination of the phaseleads to a greater uncertainty in the charge and vice versa. Chargequbits are said to operate in the charge basis (or regime), where thevalue of the charge is more localized, while phase qubits operate in thephase basis, where the value of the phase is more localized.

Charge qubits store and manipulate information in the charge states ofthe device, where elementary charges consist of pairs of electronscalled Cooper pairs. A Cooper pair has a charge of 2e, where e is theelementary charge, and consists of two electrons bound together by aphonon interaction. See, for example, Nielsen and Chuang, 2000, QuantumComputation and Quantum Information, Cambridge University Press,Cambridge, pp. 344-345.

Phase qubits, on the other hand, store information in the phase or fluxstates of the qubit. Phase qubits include a superconducting loopinterrupted by a Josephson junction. Phase qubits can be furtherdistinguished as either flux qubits or “phase-only” qubits. Flux qubitsare characterized by relatively large superconducting loops that cantrap large fluxes on the order of the unit flux Φ₀=hc/2e. See Bocko etal., 1997, IEEE Trans. Appl. Superconduct. 7 3638. “Phase-only” qubits,on the other hand, are characterized by a small inductance and aremagnetically inactive. A “phase-only” qubit stores information in theform of a phase drop across a Josephson junction interrupting thesuperconducting loop. See, for example, Ioffe et al., 1999, Nature 398,679.

Another type of qubit is the hybrid qubit. Hybrid qubits use both thecharge and phase degrees of freedom to control information. Someexamples of hybrid qubits are described in U.S. Pat. No. 6,838,694; andUnited States Patent Publication No. 2005-0082519, which are herebyincorporated by reference in their entireties.

2.4 Superconducting Flux Qubits

One proposal to build a quantum computer from superconducting qubits isBocko et al., 1997, IEEE Transactions on Applied Superconductivity 7, p.3638. See also, Makhlin et al., 2001, Review of Modern Physics 73, p.357-400. Since then, many designs have been introduced. One such designis the persistent current qubit. The persistent current qubit is a formof flux qubit, meaning that it is a phase qubit that can store fluxes onthe order of the unit flux Φ₀=hc/2e. See Mooij et al., 1999, Science285, 1036; and Orlando et al., 1999, Physics Review B 60, 15398. Asillustrated in FIG. 6, the persistent current qubit comprises a loop ofthick superconducting material interrupted by three small-capacitanceJosephson junctions (denoted as “X” in FIG. 6) in series. Thesuperconducting loop can enclose an applied magnetic flux fΦ_(O),wherein Φ_(O) is the superconducting-flux quantum h/2e, where h isPlank's constant. The value of the coefficient f can be controlled by anexternal magnetic bias and is usually kept at a value slightly smallerthan 0.5. The critical current value of one Josephson junction, denotedaE_(J) in FIG. 6, is engineered to be less than that of the criticalcurrent value E_(J) of the other two Josephson junctions, which oftenhave the same or very similar critical currents (which values are eachdenoted E_(J) in FIG. 6). Typically, a is in the range 0<a<1. Thepersistent current qubit can be built such that the loop ofsuperconducting material encloses a small area, (e.g., less than tenmicrons squared).

The persistent current qubit is well known and has demonstrated longcoherence times. See, for example, Orlando et al.; and Il'ichev et al.,2003, Physics Review Letters 91, 097906. Some other types of flux qubitscomprise superconducting loops having more or fewer than three Josephsonjunctions. See, e.g., Blatter et al., 2001, Physics Review B 63, 174511;and Friedman et al., 2000, Nature 406, 43.

The sign of the coupling interaction in the system Hamiltonian thatdescribes the coupling of two superconducting flux qubits can be used asa basis for classifying qubit coupling types. According to such aclassification scheme, there are two coupling types, ferromagnetic andanti-ferromagnetic.

Flux qubits typically interact via their respective magnetic fluxes.That is, a change in flux in a first superconducting flux qubit willcause a change in flux in a second superconducting flux qubit that iscoupled to the first superconducting flux qubit. In ferromagneticcoupling, it is energetically favorable for a change in flux of thefirst superconducting flux qubit to produce a similar change in the fluxof a second superconducting flux qubit to which the firstsuperconducting flux qubit is coupled. For example, an increase in fluxin the first qubit will cause an increase in flux in the second qubitwhen the two qubits are ferromagnetically coupled. Since circulatingloop currents generate flux within the superconducting loop of a fluxqubit, ferromagnetic coupling can also mean that circulating current inone qubit will generate current flowing in the same direction in anotherqubit.

In the anti-ferromagnetic case, it is energetically favorable for achange in flux of a first superconducting flux qubit to produce asimilar but opposite change in flux in a second superconducting fluxqubit to which the first superconducting flux qubit is coupled. Forexample, a flux increase in one qubit leads to a flux decrease in theanti-ferromagnetically coupled device. Likewise, a circulating currentin one direction in a first flux qubit causes a current flow in theopposite direction in the flux qubit that is anti-ferromagneticallycoupled to the first qubit because it is more energetically favorable.By energetically favorable, it is meant that the system comprising thecoupled qubits prefers to be in a specific coupling configuration(because the overall energy of the coupled system is lower in thespecific configuration than in other configurations).

In the Hamiltonian of two flux devices coupled together,σ_(z){circumflex over (×)}σ_(z) represents the “sigma z” couplingbetween two devices with a variable J as a pre-factor that indicates thestrength of the coupling. When J>0, the coupling is anti-ferromagnetic,with a higher J meaning a stronger anti-ferromagnetic coupling. WhenJ<0, the coupling is ferromagnetic, with a lower J meaning a strongerferromagnetic coupling. When J=0, there is no coupling. Thus, switchingthe sign of J switches the type of coupling from ferromagnetic toanti-ferromagnetic or vice versa.

2.5 Measurement Techniques for Qubits

Generally, qubit measurement is conducted based on the assumption thatthe qubit can be in a quantum state. However, qubits can be restrictedto hold only classical states and then measured when in this restrictedstate. Regardless of whether measurement relies on the assumption thatthe qubits to be measured are in a quantum state or on the assumptionthat they have been restricted to a classical state, methods andstructures in the art that can measure a large number of qubits in thesame circuit are lacking. Usually, a readout mechanism for one qubitrequires a certain amount of circuit board space, as well as at leastone control wire to operate the mechanism. Traditionally, for everyadditional qubit in a circuit, an additional readout mechanism for thatqubit is used, as well as at least one additional control wire. Thiscreates a problem in circuit design when a large number of qubits arepresent, since space constraints make placement of qubits in a circuitvery complex. Also, the presence of additional control wires creates aproblem in finding an efficient routing of all the wires in the circuit.In an array with a large number of qubits, reading out the qubits in theinterior of the array can be challenging due to restrictions in area andwiring paths into the interior of the array.

Il'ichev et al., referenced above, proposed a method to read out thestate of a flux qubit by weakly coupling the flux qubit to a tankcircuit. When the qubit is ready for measurement, the qubit is broughtinto resonance with the tank circuit so that the state of the qubit andthe state of the tank circuit couple. The tank is then decoupled fromthe qubit. This method, although it reduces dissipation of the qubit bythe tank circuit when not reading out, is not scalable to higher numbersof qubits in a quantum circuit, because having a single tank circuit foreach qubit is not feasible.

One way of measuring a flux qubit is through the use of asuperconducting quantum interference device, or SQUID, inductivelycoupled to the flux qubit. A SQUID comprises a superconducting loopinterrupted by at least one Josephson junction. The current flowing inthe loop of the SQUID can be biased in several different ways. Twoexamples of SQUIDs that differ in the way they are biased are dc-SQUIDsand rf-SQUIDs. Since flux devices interact via their magnetic fluxes, aSQUID-type device can be used to couple flux qubits together, like thescheme suggested by Majer et al., 2003, arXiv.org:cond-mat/0308192. Whenused to measure the state of a flux qubit, the SQUID's supercurrent isread out because this supercurrent is dependent on the state of thequbit. As such, a measurement of the SQUID's current can determine thestate of the qubit to which the SQUID is coupled. However, SQUIDs havethe drawback that they take up a considerable amount of surface area ona circuit board or chip. For higher numbers of qubits, having a SQUIDfor each qubit becomes cumbersome and space consuming.

Paternostro et al., 2005, Physical Review A 71, 042311, (hereinafter“Paternostro”) disclose a method of transferring a quantum state of aqubit through a multi-qubit coupling via a bus system. Paternostrocombines quantum optics and SQUIDs in order to create a network of spinchains on which quantum operations can be performed. However, includinga bus to couple all the qubits together can introduce increased noiseinterference into the system.

2.6 State of the Art

Given the above background, there exists a need in the art to providesystems and methods for efficiently reading out the classical state ofqubits in an array, especially qubits in the interior of the array.

3. BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates the two circulating current states of an rf-SQUID.

FIG. 1B illustrates the energy landscape of the rf-SQUID in FIG. 1A.

FIG. 1C illustrates a split junction flux qubit, in accordance with theprior art.

FIG. 2 illustrates two flux qubits and a coupling device in accordancewith an embodiment of the present methods and systems.

FIG. 3A illustrates sequential steps involved in ferromagnetic statecopying in accordance with an embodiment of the present methods andsystems.

FIG. 3B illustrates sequential steps that follow the steps of FIG. 3Ainvolved in ferromagnetic state copying in accordance with an embodimentof the present methods and systems.

FIG. 4A illustrates sequential steps involved in adiabatic state copyingin accordance with an embodiment of the present methods and systems.

FIG. 4B illustrates sequential steps that follow the steps of FIG. 4Ainvolved in adiabatic state copying in accordance with an embodiment ofthe present methods and systems.

FIG. 5 illustrates an array of coupled flux qubits with readout devicesaround the periphery in accordance with an embodiment of the presentmethods and systems.

FIG. 6 illustrates a persistent-current qubit in accordance with theprior art.

4. SUMMARY OF THE INVENTION

In one embodiment, a method of copying a classical state of a firstqubit to a second qubit is provided. The method comprises initializingthe second qubit to an initial classical state. The second qubit has apotential energy configuration comprising a first potential well havinga first potential minimum and a second potential well having a secondpotential minimum. The initial classical state is located in the firstpotential well. In the method the first potential minimum of the firstpotential well is adjusted to a third potential minimum that is higherthan the second potential minimum of the second potential well. Themethod further comprises coupling the first qubit and the second qubitfor a duration t.

In another embodiment, a method of copying a classical state of a firstqubit to a second qubit is provided. The first qubit is characterized bya potential energy configuration that comprises a first tunnelingbarrier, and the second qubit is characterized by a potential energyconfiguration that comprises a second tunneling barrier. The methodcomprises lowering the second tunneling barrier, coupling the firstqubit and the second qubit for a duration t, and raising the secondtunneling barrier.

In yet another embodiment, a method for reading out a classical state ofa qubit in an array of qubits is provided. The array comprises perimeterqubits and interior qubits. The method comprises initializing aclassical state of a perimeter qubit having an associated readoutdevice, copying a classical state of an interior qubit to the perimeterqubit, and reading out the classical state of the interior qubit byreading out the classical state of the perimeter qubit. The perimeterqubit is coupled to the interior qubit via a coupling device having acoupling strength. Further, the coupling strength is adjustable betweena minimum coupling strength and a predetermined coupling strength.

In still another embodiment, a method of copying a classical state of afirst qubit means to a second qubit means is provided. The methodcomprises means for coupling the first qubit means to the second qubitmeans, means for adjusting at least one of a tunneling barrier of thefirst qubit means and a tunneling barrier of the second qubit means, andmeans for adjusting a symmetry of a potential energy configuration of atleast one of the first qubit means and the second qubit means.

In still another embodiment, a system for copying a classical state of afirst qubit to a second qubit is provided. The first qubit ischaracterized by a potential energy configuration that comprises a firsttunneling barrier, and the second qubit is characterized by a potentialenergy configuration that comprises a second tunneling barrier. Thesystem comprises a first barrier adjustment module, a coupling module,and a second barrier adjustment module. The first barrier adjustmentmodule comprises instructions for lowering the second tunneling barrier.The coupling module comprises instructions for coupling the first qubitto the second qubit. The second barrier adjustment module comprisesinstructions for raising the second tunneling barrier.

In still another embodiment, a computer-readable medium storingexecutable instructions for initializing a first qubit to an initialclassical state is provided. In this embodiment, the first qubit has apotential energy configuration comprising a first potential well havinga first potential minimum and a second potential well having a secondpotential minimum, and the initial classical state is located in thefirst potential well. The computer-readable medium further storesexecutable instructions for adjusting the first potential minimum of thefirst potential well to a third potential minimum that is higher thanthe second potential minimum of the second potential well. Thecomputer-readable medium further stores executable instructions forcoupling the first qubit and the second qubit for a duration t.

5. DETAILED DESCRIPTION

As will be described in further detail below, the present methods andsystems provide for copying the classical state of a first qubit to asecond qubit. In some embodiments, the first and second qubits arecoupled and the escape probability of the second qubit is tuned. Inother embodiments, the tunneling barrier of the second qubit isinitialized to a high value and decreased, the qubits are coupled, andthen the tunneling barrier of the second qubit is raised to copy thestate of the first qubit.

The present methods and systems may also provide for reading out thestates of an array of qubits. In some embodiments, the array may betwo-dimensional, with the qubits in the outer perimeter of the arraybeing read out using techniques known in the art. The states of qubitsadjacent to the perimeter qubits are then copied using the presentmethods and systems to corresponding adjacent qubits in the outerperimeter of the array. Once copied, the states are read out usingtechniques known in the art, thereby providing a mechanism for readingout the state of qubits in the interior of the array. In some instances,this process continues with qubits increasingly deeper in the interiorof the array until the entire array has been read out. Readout of qubitsin the interior of the array may be done multiple times to increaseaccuracy of measurement.

Qubits, such as flux qubits, function as two-level systems. That is, aqubit has two distinct states that hold information. For example, anrf-SQUID 100A, which can be used as a flux qubit, is shown in FIG. 1A.The rf-SQUID 100A comprises a main superconducting loop 103 interruptedby Josephson junction 101. The two distinct states of rf-SQUID 100A arethe two directions of circulating current around the loop, respectivelyshown as arrows 102-0 and 102-1. rf-SQUID 100A can be in either aclassical state, where current is flowing in one direction only in thesuperconducting loop, or in a quantum superposition of states, wherecurrent is flowing in both directions at the same time in thesuperconducting loop. FIG. 1B shows the corresponding energy diagram forrf-SQUID 100A. The potential energy landscape 100B is a bistablepotential with two minima 160-0 and 160-1 and an energy barrier 140.Minima 160-0 and 160-1 can be degenerate, meaning that they have thesame energy, in some instances. In other instances, minima 160-0 and160-1 are not degenerate. When the minima are degenerate, the energylandscape is referred to as symmetric. Current directions 102-0 and102-1 in FIG. 1A respectively correspond to potential wells 160-0 and160-1 in the minima of FIG. 1B. However, this specific correspondence isarbitrary. Using this correspondence, a qubit having the classical statecorresponding to current 102-0 of FIG. 1A is said to be located in theleft potential well, well 160-0 of FIG. 1B. Similarly, a qubit havingthe classical state corresponding to current 102-1 of FIG. 1A is said tobe located in the right potential well, well 160-1 of FIG. 1B.

The state of qubit 100A can tunnel quantum mechanically through energybarrier 140 from one minimum to the other. The frequency of thistunneling depends on the height of the barrier. If the barrier is high,less tunneling occurs. If the barrier is low, tunneling occurs moreoften. When little or no tunneling occurs (high barrier), the qubit issaid to be in the classical regime. When the tunneling rate is high (lowbarrier), the qubit is said to be in the quantum regime. When a qubitholds a classical state, it is meant that the state of the qubit isentirely localized in one well and is not in a superposition of bothwells.

Energy landscape 100B can be tuned by changing the bias parameters ofrf-SQUID 100A. For example, tuning the Josephson energy of the qubit canchange the height of the energy barrier 140. To be able to tune theJosephson energy of junction 101, two Josephson junctions in a smallloop, known as a split junction, can replace single junction 101. Anexample of a split junction flux qubit 100C is shown in FIG. 1C. Itcomprises two Josephson junctions 101-1 and 101-2 that form a smallsuperconducting loop 104 in addition to main loop 103. Tuning themagnetic flux in small loop 104 of the split junction changes theeffective Josephson energy of the split junction. One can also tune theJosephson energy of junction 101 by applying a transverse (in-plane)magnetic flux through Josephson junction 101. The potential minima ofwells 160-0 and 160-1 can be changed relative to one another byadjusting the magnitude of the magnetic flux Φ_(x) that is applied tomain loop 103. To make such an adjustment, magnetic field flux can beapplied by an inductive bias loop (not shown) that is proximate to qubit100C. The strength of the magnetic moment produced by the circulatingcurrent in main loop 103 can be tuned by changing the Josephson energyof Josephson junction 101, which is done using methods such as thosedescribed above. These features allow greater flexibility in thebehavior of rf-SQUID 100C.

Initializing a flux qubit means that the qubit is put into a known stateusing, for example, initialization methods known in the art. This isusually done before the start of a quantum operation involving thequbit. To initialize an rf-SQUID, such as 100A illustrated in FIG. 1A,to a classical state, the state of the qubit is localized to one of itspotential wells (e.g., well 160-0 or well 160-1). One way to accomplishthis is to make the energy landscape highly non-degenerate. For example,if the qubit is to be initialized to potential well 160-1, the energyminimum of potential well 160-0 is raised to a value slightly lower thanthe height of energy barrier 140, thus making well 160-0 “shallow”compared to well 160-1. Adjusting the amount of flux applied to thesuperconducting loop of the qubit can raise the potential energy minimumof well 160-0. Well 160-0 can be made to be only slightly lower thanenergy barrier 140. As used here, in some instances, the term “slightlylower” means that the value of the potential energy minimum of well160-0 is within about eighty to ninety percent of the value of energybarrier 140. In such an unstable state, there is a high probability thattunneling will occur and that the state of the qubit will be localizedto well 160-1. Note that, in such a scenario, the potential energy ofwell 160-1 is much lower than energy barrier 140, so no tunneling willoccur out of well 160-1. Once the state of rf-SQUID 100A has beenlocalized to well 160-1, well 160-0 is lowered to about its originalvalue.

Methods for reading out the state of flux qubits like rf-SQUID 100A arewell known in the art. However, readout schemes proposed so far areuseful only for a small number of qubits and are not scalable. If alarge number of qubits were present in a circuit, the space and wiringconstraints would render these schemes ineffective. Since it ispreferable to be able to read out the state of any qubit in a quantumcomputer or quantum processor, systems in which each qubit has anassociated readout device are desired.

One characteristic of the present methods and systems is the absence ofany requirement that each qubit in a group of qubits have a readoutdevice for the states of all qubits to be measured. If the states of thequbits were classical, which can be achieved by raising the tunnelingbarrier 140 between the two states of the qubit as described above, thenthe states of qubits without readout devices can be copied to qubitsthat do have readout devices. Such a technique does not violate thequantum “no-copy” rule, which states that a quantum state cannot becopied exactly. Since the qubit is in a classical state, meaning that itis not a superposition of two states, there is no physical obstacle thatprevents the copying of the state. Aspects of the present methods andsystems include two techniques for inductively copying the classicalstate from one qubit to another.

Ferromagnetic State Copying

FIG. 2 shows two rf-SQUID flux qubits 201 and 203 and an rf-SQUIDcoupling device 202 that can couple the qubits ferromagnetically oranti-ferromagnetically, or tune the coupling close to zero. Otherinductive coupling devices, like dc-SQUIDs or flux transformers, can beused in place of rf-SQUID 202, as long as the device still fulfills theattributes of the coupling device described herein. Coupling device 202may provide tunable coupling, and may include a split junction.Likewise, other types of flux qubits, like the persistent current qubitof Orlando et al., 1999, Physics Review B 60, 15398, which is herebyincorporated herein by reference can be used in place of rf-SQUIDs 201and 203. Qubits 201 and 203 may each comprise an rf-SQUID with a splitjunction.

The ferromagnetic state copying method for copying the classical stateof qubit 201 to qubit 203 is illustrated in FIGS. 3A and 3B, with thesteps of FIG. 3B sequentially following the steps of FIG. 3A. FIGS. 3Aand 3B illustrate the energy landscape of qubits 201 and 203 as well asthe state of these qubits at different points in the method. The energylandscape of both qubits 201 and 203 is a bistable potential, whichcomprises two potential minimum wells (the left and right wells shown ineach of the potential energy diagrams of FIGS. 3A and 3B).

Step 301. Step 301 shows the initial condition of the qubits, which isusually at the end of a calculation or evolution. The states of qubitsare not known, and thus the state of the qubits is represented byunfilled circles in both minima of the energy landscape of both qubits.Energy barrier 140 between the minima may or may not be low enough toallow quantum tunneling between the minima. Qubits 201 and 203 may bedegenerate, or nearly degenerate.

Optional step 302. When the state of qubit 201 is to be copied to qubit203, the energy barrier 140 of both qubits may be raised to a highenough value to prevent quantum tunneling from occurring, effectivelyprojecting the state of the qubits to one of the classical basis states.This is illustrated in step 302. If the energy barriers are already highenough to prevent tunneling at the end of step 301, step 302 can beomitted. Black circles used throughout FIGS. 3A and 3B denote thelocalization of the state of a qubit to a single well. Thus, the blackcircle in qubit 201 in step 302 indicates that the state of the qubit islocalized in the right well. This choice of localization to the rightwell as opposed to localization to the left well in the illustration ofstep 302 of the inventive method is arbitrary and only for illustrativepurposes. In practice, the identity of the well to which the state ofthe qubit is localized will depend on the quantum operations or timeevolution performed on the qubit before step 301. Raising energy barrier140 of a qubit can be achieved by tuning the Josephson energy of thequbit, for example, using any of the techniques that have beenpreviously described. The patterned circles in qubit 203 in step 302indicate that the state of qubit 203 is located in one of the wells, butwhich well it is in is not significant.

Step 303. After energy barriers 140 have been raised in optional step302, any couplings between qubits 201 or 203 and other devices arereduced to zero or near zero in order to prevent further interactionsthat might change the state of the qubits. Reducing the coupling to zeroor near zero may be done concurrently with the raising of energybarriers 140. Thus, steps 302 and 303 may be done at the same time.There may be an initial ferromagnetic coupling between qubits 201 and203 and it is the only coupling not reduced to zero (or close to zero)during step 303. In step 303, qubit 203 is arbitrary initialized to oneof the wells. This can be done, as described previously, by raising oneof the potential wells, thereby localizing the state of the qubit to theother well that has lower potential energy. This raising step is notillustrated in FIG. 3A. As illustrated in FIG. 3A, the state of qubit203 was initialized to the left well, but this choice is arbitrary. Thequbit could have been initialized in the right well.

Step 304. In step 304, the potential minimum of the well that qubit 203is initialized to is raised. In this case, the left well is raised. Inother words, the minimum potential energy of the left well is raised sothat the bistable potential of qubit 203 has a shallow well (left) and adeep well (right). Raising the minimum potential energy in the left wellto a value close to but less than the energy barrier height 140increases the probability that qubit 203 will tunnel through the barrierfrom the left well into the right well. However, tunneling from theright well to the left well is highly improbable. Changing the depth ofthe well can be achieved by tuning the magnetic flux through thesuperconducting loop of the qubit. If qubit 203 were initialized to theright well then, in step 304, the potential minimum of the right wellwould be raised.

Optional step 305. In step 305, a strong ferromagnetic coupling 330 isintroduced between qubits 201 and 203 if there is no coupling alreadypresent. Coupling 330 can be introduced by turning on a coupling devicebetween qubits 201 and 203, like rf-SQUID coupling device 202 in FIG. 2.In some cases, a strong coupling may be defined as a maximum potentialcoupling strength that coupling device 202 can achieve, and a strongferromagnetic coupling may be defined as a coupling strength that isequivalent to J=−1 in the Hamiltonian of the coupled system, where J isthe pre-factor of the qubit coupling term. In the case where aferromagnetic coupling is already present between qubits 201 and qubit203 before step 305, for example when coupling device 202 is already on,step 305 can be omitted. When a ferromagnetic coupling between thequbits is applied, it is energetically favorable for the states of bothqubits to be the same. Since the energy barrier of qubit 201 is high,qubit 201 cannot change states. However, the state of qubit 203 isheavily influenced by the state of qubit 201 through ferromagneticcoupling 330.

Step 306. In the illustrated example, the state of qubit 201 is in theright well and the state of qubit 203 is in the shallow left well.Therefore, qubit 203 will tunnel through the energy barrier into thelower right well in order to reduce the energy of the coupled system, asshown in step 306. This is because ferromagnetic coupling 330 causes itto be energetically more favorable for both qubits to hold the samestate. Since tunneling in qubit 201 cannot occur due to its hightunneling barrier, only qubit 203 is able to change its state in orderto match the state of qubit 201. If, on the other hand, the state ofqubit 201 was in the left well (not shown), qubit 203 would not tunnelbut instead would stay in the left well with a large probability (notshown). There is a small probability that qubit 203 would still tunnelin this case. To eliminate copying errors due to errant tunneling,multiple copy operations can be performed and averaged.

Optional step 307. Ferromagnetic coupling 330 is removed in optionalstep 307. Alternatively, ferromagnetic coupling 330 may not be removed,in which case step 307 is omitted.

Step 308. The minimum of the left well of qubit 203 is lowered (e.g., toits original value) in step 308. This is achieved by biasing themagnetic flux through a loop of qubit 203. Thus, at the end of step 308,the state of qubit 203 is the same as the state of qubit 201. Theclassical state of qubit 201 has been copied to qubit 203. The timeinterval in which steps 304 and 308 are completed (that is, the processof raising and lowering the potential) is called the tipping time. Thoseof ordinary skill in the art will appreciate that the selection of asuitable tipping time may vary. In some cases, the tipping time may bebetween about 1 ns and 800 μs, e.g. 10 μs.

By way of illustration, a numerical example of the strength of couplingis presented. Qubits 201 and 203 are rf-SQUIDs, each with a loop size of25 μm×25 μm, a loop inductance of 50 pH, and characterized by a criticalcurrent of 15 μA. Coupling device 202 is an rf-SQUID with a loop size of25 μm×25 μm, a loop inductance of 50 pH, and a critical current of 5.5μA. The mutual inductance between qubit 201 and coupling device 202 isapproximately 3 pH and the spacing between them is about 1 μm. Themutual inductance between qubit 203 and coupling device 202 isapproximately 3 pH and the spacing between them is about 1 μm.

Let the magnetic flux applied to the superconducting loop of both qubitsbe kept at Φ₀/2, where Φ₀ is the magnetic flux quantum. When the qubitsare biased at this amount, the difference in circulating current betweentheir classical states is around 26 μA. Coupler 202 is turned on andbiased to provide ferromagnetic coupling between the qubits. When thestate of one qubit (say 201) is switched, this produces ananti-ferromagnetic coupling between qubit 201 and coupling device 202.The change in circulating current in the coupler due to qubit 201switching states is around 7 μA. This corresponds to a change in flux inthe coupler of 0.038Φ₀. Likewise, this change in flux in the couplingdevice produces a change in flux in the other qubit (203) of around0.01Φ₀ due to anti-ferromagnetic coupling. Although the qubits arecoupled anti-ferromagnetically to the coupling device, the qubit-qubitcoupling mediated by the coupling device is ferromagnetic in nature.This coupling is strong enough such that the state copying fidelity of acopying operation is greater than 99.999% at sufficiently lowtemperature (<100 mK). The spin copying operation can be performedmultiple times to increase the overall copying fidelity, if required.

Adiabatic State Copying

An embodiment of the present methods and systems for adiabaticallycopying the classical state of one qubit to another, like from qubit 201to qubit 203 in FIG. 2, is illustrated in FIGS. 4A and 4B, with thesteps in FIG. 4B sequentially following the steps of FIG. 4A. The energylandscape of both qubits is a bistable potential, like the oneillustrated in FIG. 1B, and may or may not be degenerate.

Step 401. Step 401 shows the initial condition of the qubits, which isusually at the end of a calculation or evolution. There is norequirement that the state of the qubits be known in step 401. Thus, thestate of qubits 201 and 203 is represented by unfilled circles in bothminima of the bistable potential for both qubits in FIG. 4A. The energybarrier between the minima may or may not be low enough to allow quantumtunneling between the minima.

Optional step 402. If the state of qubit 201 is to be copied into qubit203, the energy barrier of both qubits is raised to a high enough valueto prevent quantum tunneling from occurring, effectively projecting thestate of the qubits to one of the classical basis states. This isillustrated in step 402. If the energy barriers are already high enoughto prevent tunneling, then step 402 can be omitted. The black circle inqubit 201 in step 402 represents the state of qubit 201 as beinglocalized in the right well. This choice is arbitrary and only forillustrative purposes. In practice, the state of qubit 201 will dependon the calculation or time evolution performed on it prior to step 401.Raising the energy barrier in each qubit can be achieved by tuning theJosephson energy of qubit 201 and of qubit 203. In FIG. 4A, qubit 203 islocalized in the left well, but this choice is only for illustrativepurposes. In practice, the state of qubit 203 can be in either well atthe end of step 402. Both qubits are biased with a flux equal to half aflux quantum (e.g. the qubits are within their hysteresis region, thatis, the region where changing the qubit's flux bias does not cause achange of its classical state), and this bias is maintained throughoutthe process. After the barriers are raised, any couplings between qubit201 or 203 and other devices are reduced to zero (or close to zero). Thereduction of the coupling to external devices to zero (or close to zero)and the raising of the energy barriers may be done concurrently. In somecases, there may be an initial ferromagnetic coupling between qubits 201and 203 and it is the only coupling not reduced to zero (or close tozero).

Step 403. In step 403, the energy barrier of qubit 203 is adiabaticallyreduced to bring the qubit from the classical regime to the quantumregime. Reducing the height of the barrier can be achieved by tuning amagnetic field transverse to qubit 203 or by tuning the Josephson energyof the qubit. The minimum amount of time needed to achieve step 403 isthe amount of time needed to sufficiently guarantee that unwantedtunneling events will not be induced. Those of ordinary skill in the artwill appreciate that the time needed for step 403 will vary. Forexample, in some cases it may be between about 1 ns and 1 ms, e.g. 100μs. The height of the barrier may be reduced to zero or near zero. Oncequbit 203 is brought into the quantum regime, tunneling between thepotential wells of the qubit can occur.

Step 404. A strong ferromagnetic coupling 430 is introduced betweenqubits 201 and 203 in step 404. In some cases, a strong ferromagneticcoupling may be defined as a coupling strength that is equivalent toJ=−1 in the Hamiltonian of the coupled system, where J is the pre-factorof the qubit coupling term. In the case where no coupling exists betweenqubits 201 and 203 prior to step 404, step 404 comprises turning oncoupling device 202. In the case where a ferromagnetic coupling isalready present between qubits 201 and qubit 203 before step 404(coupling device 202 is already on), then step 404 can be omitted. Whena ferromagnetic coupling is applied, it is energetically favorable forthe states of both qubits to be the same. Since energy barrier 140 ofqubit 201 is high, qubit 201 cannot change states. However, since theenergy barrier of qubit 203 is low, it is possible for the state ofqubit 203 to tunnel from one well to the other. Thus, if the state ofqubit 203 is in the left well (and the state of qubit 201 is in theright well as depicted in step 404 of FIG. 4A), the coupling would causethe qubit to tunnel into the right well. If the state of qubit 203 is inthe right well in step 404 (not shown), then no tunneling would occursince the qubit is already in the energetically favorable state. Themethod would work in a similar fashion if the state of qubit 201 was inthe left well (not shown), with the state of qubit 203 tunneling intothe left well if not already in the left well.

Step 405. In step 405, it is shown that qubit 203 has localized in thesame state as qubit 201 due to ferromagnetic coupling 430.

Step 406. In step 406, the energy barrier of qubit 203 is adiabaticallyreturned to a level that reduces the amount of quantum tunnelingpermitted, thus transitioning the qubit from the quantum regime backinto the classical regime. This prevents qubit 203 from tunneling out ofthe state it was at the end of step 405. Also, ferromagnetic coupling430 can be removed if desired. The time taken to complete step 406 islong enough so as to not induce unwanted tunneling events, and may be ofthe same order as step 403.

Step 407. In step 407, qubit 203 is back in the classical regime and hasthe same state as qubit 201. A flux bias of half a flux quantum ismaintained on both qubits during the entire copying operation (steps 401to 407). The flux bias is within a certain tolerance in order for thecopying operation to succeed, and may be equal to the amount of fluxcoupled from qubit 201 to qubit 203. For example, if the amount of fluxcoupled is 0.01Φ₀, then the accuracy needed for the qubit flux bias maybe greater than ±0.01Φ₀.

Readout of an Array of Flux Qubits

FIG. 5 shows a two-dimensional array 200 of flux qubits 510. Flux qubits510 in the interior of the array are labeled B and are coupled to fournearest neighbors by coupling devices 520. Flux qubits 510 on theperimeter of the array are labeled A and are coupled to two or threeadjacent qubits, depending on whether the qubit is located on a corneror an edge. Although it is not shown, each qubit 510 may also be coupledto one or more of its four next-nearest neighbor qubits throughadditional coupling devices aligned diagonally in the array. Inconventional qubit arrays, each qubit has an associated readout devicethat can measure the state of each qubit. However, if the array islarge, having a readout device for each qubit quickly becomescumbersome. Even for the 4×4 array shown in FIG. 5, having a readoutdevice for each qubit becomes a complex design problem, especially forqubits (B) in the interior of the array. Even in the case where couplingdevices that couple the qubits together are used as readout devices, thedesign complexity is still undesirably high.

An aspect of the present methods and systems is the application offerromagnetic state copying or adiabatic state copying to system 500. Bycopying the classical state of the interior qubits (B) to the perimeterqubits (A), the interior qubits (B) can be read out without having aspecific readout device associated with it. A perimeter qubit (A) ishereinafter defined as a qubit that has an associated readout device.Perimeter qubits (A) are usually located around the periphery of thearray of qubits as illustrated in FIG. 5. However, as the termed is usedherein, there is no absolute requirement that perimeter qubits (A) be onthe perimeter of the array. As such, an interior qubit (B) ishereinafter defined as a qubit in the array that does not have anassociated readout device. Thus, a qubit that does not have anassociated readout device is an interior qubit (B) even if it is locatedon the periphery of the array. Readout devices 540 may be placed aroundthe periphery to reduce the layout complexity of the array.

As an example, the state of qubit 510-2 (B) can be copied to qubit 510-1(A), and then read out by one of the readout devices 540-1. Couplingdevices 520 are capable of ferromagnetically coupling qubits togetherand are capable of turning the coupling off. Coupling devices 520 mayalso be capable of anti-ferromagnetically coupling two qubits together.Coupling devices 520 may be similar to coupling devices 202 describedherein. The coupling strength of coupling devices 520 may becontrollably tunable. Flux qubits 510 may be rf-SQUIDs, persistentcurrent qubits, or any other type of qubit that stores information inits flux states. Various readout devices 540 are well known in the art,such as dc-SQUIDs.

A method for reading out the classical state of interior qubit 510-2 (B)is now described. In some cases, all coupling devices 520 in array 500that are coupled to qubit 510-2 may be turned off. This ensures that thestates of the other qubits do not mix with the state of qubit 510-2.Alternatively, coupling devices 520 coupled to qubit 510-2 may be on andthe states of the qubits coupled to qubit 510-2 are all known. In thiscase, even though the state of qubit 510-2 is mixed with the states ofother qubits, one can determine what the state of qubit 510-2 is becauseall the other states are known.

The state of an adjacent qubit that is a perimeter qubit, qubit 510-1,for example, is first read out by one of the readout devices 540-1. Thisreadout may be done more than once, or even multiple times, to increasethe fidelity of measurement. Once the information of qubit 510-1 hasbeen obtained, the state of the qubit can be initialized to an arbitrarystate. Next, the classical state of qubit 510-2 is copied to qubit 510-1using coupling device 520-1. The techniques for classical state copying,specifically ferromagnetic state copying (FIG. 3) and adiabatic statecopying (FIG. 4), have been described previously for the system in FIG.2. These techniques can be applied in the same way to qubits in thearray of FIG. 5. Once copying has finished, qubit 510-1 will have thesame state as qubit 510-2. Qubit 510-1 is then read out by one of thereadout devices 540-1, thus effectively reading out the state of qubit510-2. The state of qubit 510-2 can be copied to qubit 510-1 and readout multiple times to increase measurement fidelity.

Qubit 510-2 is not limited to copying its state to qubit 510-1. Thestate of qubit 510-2 can be copied to any qubit it is coupled with,provided that the state of the qubit it copies to is either alreadyknown (e.g. has been read out already) or is not needed so thatinformation is not lost. The state of qubit 510-2 can be copied andpropagated in any direction until it reaches a perimeter qubit, at whichpoint the state is measured. The path that the copied state of qubit510-2 takes to reach a perimeter qubit may be the shortest pathpossible. For example, the shortest path for qubit 510-2 to copy iseither the qubit to the left (qubit 510-1) or the qubit to the top,since both are perimeter qubits. Qubits to the right and to the bottomof qubit 510-2 are also interior qubits, so the path to a perimeterqubit is longer in this case.

Readout devices 540 may be placed on all sides of the two-dimensionalarray and there is a readout device for every qubit on the periphery ofthe array, as illustrated in FIG. 5, or may be placed on only some sidesof the array. For example, readout devices 540-1 and 540-3 may bepresent while 540-2 and 540-4 are not. Alternatively, there may be onlyone readout device on the perimeter of the array. The number of readoutdevices available affects the copy path taken from the qubit to becopied to a perimeter qubit that has a readout device as well as thenecessity of turning off couplings to the qubit that is to be copied.This assumes that the states of all the qubits in the copy path arealready known or are not needed.

FIG. 5 shows a 4×4 array, but the concept easily scales to larger arraysizes. The readout procedure for larger arrays is similar to what wasdescribed for a 4×4 array. That is, the perimeter qubits are read outfirst and then their states are reset. Next, the states of interiorqubits are copied to perimeter qubits and read out in the mannerdescribed above. There is no requirement that all the perimeter qubitsbe readout before any of the interior qubits are read out. All that isrequired is that a given perimeter qubit be read out prior to using theperimeter qubit to readout the state of an interior qubit. Thus, it ispossible for some interior qubits to be readout before each of theperimeter qubits are readout. Copy and readout operations can be donemultiple times to increase measurement fidelity. For instance, the samecopy and readout operations can be done twice, three times, four times,five times, more than five times, more than ten times, or more than 100times.

Coupling devices between qubits may couple the qubits togetherferromagnetically during state copying, so that the state of the qubitbeing copied to is the same as the state of the qubit being copied.Alternatively, coupling devices between qubits may couple the qubitstogether either ferromagnetically or anti-ferromagnetically during statecopying. Anti-ferromagnetic coupling has the effect that the qubit beingcopied to has the opposite state as the qubit being copied. In somecases where anti-ferromagnetic coupling is used, there may be an evennumber of anti-ferromagnetic state copying operations between the qubitbeing copied and the perimeter qubit that is being read out, such thatthe state of the perimeter qubit is the same as that of the qubit beingcopied. Alternatively, if there are an odd number of anti-ferromagneticcopies, then the perimeter qubit will have the opposite state as thequbit being copied and the fact that the state being read out isopposite to the state of the qubit being copied may be compensated forin post-processing.

Device Design and Parameters

Device designs and parameters are proposed for the system of FIG. 2 thatwould suit the present methods and systems for state copying. Qubits 201and 203 may be rf-SQUIDs, or persistent current qubits. In some cases,qubits 201 and 203 may have a loop area between about 5 μm² and 100000μm². For example, the loop area of qubits 201 and 203 may be square andhave dimensions of 25 μm×25 μm (625 μm²). For a qubit of this size, theloop inductance is approximately 50 pH. The critical current of qubits201 and 203 may be between about 0.1 μA and 300 μA, e.g. 15 μA. In somecases, the critical current, loop inductance, and loop area of qubits201 and 203 may be close but not the same.

The Josephson energy, and therefore the critical current, of theJosephson junction in qubits 201 and 203 may be tunable. TunableJosephson energy may be achieved by replacing the single Josephsonjunction with two parallel junctions that form a loop, otherwise knownas a split junction. An example of a split junction flux qubit is shownin FIG. 1C. When the qubits have a split junction, tuning the fluxthrough the loop of the split junction changes the effective Josephsonenergy of the split junction, which also changes the critical current.Tuning the Josephson junction also has the effect of changing the heightof the energy barrier in a qubit. Tunable Josephson energy can also beachieved by applying an in-plane magnetic field through the junction.

In some cases, coupling device 202 may have a loop area between about 5μm² and 1000 μm². For example, coupling device 202 may be an rf-SQUID,with a square loop area of dimensions of 25 μm×25 μm (625 μm²), and aloop inductance of approximately 50 pH. In some cases, the criticalcurrent of coupling device 202 may be between about 1 μA and 10 μA, e.g.about 5.5 μA. Coupling device 202 may have a split junction and itsJosephson energy may be tunable.

In some cases, the mutual inductance between qubits 201 and 203 andcoupling device 202 may be between about 0.5 pH and 20 pH, e.g. 3 pH.The mutual inductance is determined by the geometry of the devices andthe distances between them. In some cases, the spacing between each ofqubit 201 and 203 and coupling device 202 may be between about 0.1 μmand 10 μm, e.g. 1 μm. A portion of the loop of either qubit 201 or 203may overlap a portion of the loop of coupling device 202. For example, aportion of wire from a loop of one device may be placed on top (onanother layer) of a portion of wire from a loop of another device. Thereis no galvanic contact between the devices. Overlapping wires increasesthe mutual inductance between two devices, and therefore increases thecoupling strength.

In some cases, the temperature at which system 200 and/or 500 operatesat is between about 1 mK and 4 K. For example, the temperature at whichsystem 200 and/or 500 operates may be about 500 mK, which is close tothe macroscopic quantum tunneling crossover temperature.

Anti-Ferromagnetic State Copying

Embodiments described above for copying a classical state of one qubitto another qubit described involve ferromagnetic coupling between thequbits. However, it will be clear to those of ordinary skill in the artthat in the present methods and systems, qubits may be coupled togetheranti-ferromagnetically. Such coupling is referred to herein asanti-ferromagnetic state copying. Although the term “anti-ferromagneticstate copying” is used it will be understood that the classical state isnot literally copied. Rather, the target qubit to which the state of anoriginating qubit is copied using the anti-ferromagnetic state copyingmethod adopts a state that is opposite to that of the originating qubit.The techniques for anti-ferromagnetic state copying are similar to thosedescribed for ferromagnetic state copying except for the fact that thecoupling between the qubits is anti-ferromagnetic, not ferromagnetic andthe target qubit has a state that is opposite that of the originatingqubit. In some cases, some qubit pairs in a given array areferromagnetically coupled while others are anti-ferromagneticallycoupled. All that is required in such topologies is correct bookkeepingof the coupling types between the originating qubit and the targetqubit, so that a determination can be made as to whether to reverse thestate of the target qubit upon readout or not in order to achieve arepresentation of the state of the originating qubit.

CONCLUSION AND REFERENCES CITED

As will be apparent to those skilled in the art, the various embodimentsdescribed above can be combined to provide further embodiments. Aspectsof the present systems, methods and apparatus can be modified, ifnecessary, to employ systems, methods, apparatus and concepts of thevarious patents, applications and publications to provide yet furtherembodiments of the present methods and systems. As used herein, the term“about” means within approximately ±5 to 20% of the stated value.Furthermore, variations in the fabrication details of devices 201, 202,and 203 are within the scope of the present invention. These and otherchanges can be made to the present systems, methods and apparatus inlight of the above description. In general, in the following claims, theterms used should not be construed to limit the invention to thespecific embodiments disclosed in the specification and the claims, butshould be construed to include all possible embodiments along with thefull scope of equivalents to which such claims are entitled.Accordingly, the invention is not limited by the disclosure, but insteadits scope is to be determined entirely by the following claims.

1. A method of copying a classical state of a first qubit to a second qubit, the method comprising: initializing the second qubit to an initial classical state, wherein the second qubit has a potential energy configuration comprising a first potential well having a first potential minimum and a second potential well having a second potential minimum, and the initial classical state is located in the first potential well; adjusting the first potential minimum of the first potential well to a third potential minimum that is higher than the second potential minimum of the second potential well; and coupling the first qubit and the second qubit for a duration t.
 2. The method of claim 1, wherein the coupling step comprises ferromagnetically coupling the first qubit and the second qubit.
 3. The method of claim 1, wherein the coupling step comprises anti-ferromagnetically coupling the first qubit and the second qubit.
 4. The method of claim 1, wherein the first qubit has a potential energy configuration that comprises a first tunneling barrier and the potential energy configuration of the second qubit comprises a second tunneling barrier; the method further comprising raising at least one of the first tunneling barrier and the second tunneling barrier prior to the initializing step.
 5. The method of claim 4, wherein the third potential minimum is at least slightly lower than the second tunneling barrier.
 6. The method of claim 4, wherein the raising step comprises applying a transverse magnetic field to at least one of the first qubit and to the second qubit.
 7. The method of claim 4, wherein the raising step comprises tuning a Josephson energy of at least one of the first qubit and the second qubit.
 8. The method of claim 1, further comprising adjusting the first potential well from the third potential minimum back to approximately the first potential minimum after the coupling step.
 9. The method of claim 1, further comprising reducing a coupling strength between the first qubit and a device coupled to the first qubit to about zero prior to the initializing step.
 10. The method of claim 1, further comprising reducing a coupling strength between the second qubit and a device coupled to the second qubit to about zero prior to the initializing step.
 11. The method of claim 1, wherein the first potential minimum is about equal to the second potential minimum.
 12. The method of claim 1, wherein a potential energy configuration of the first qubit is a bistable potential during the initializing, adjusting, and coupling steps.
 13. The method of claim 1, wherein the duration t is between about 1 μs and 100 μs.
 14. The method of claim 1, wherein the duration t is greater than about 1 ns.
 15. The method of claim 1, wherein the coupling step results in a final classical state of the second qubit matching the classical state of the first qubit.
 16. The method of claim 15, wherein the final classical state is reached by a tunneling from the initial classical state to the final classical state.
 17. The method of claim 1, wherein the second qubit comprises a superconducting loop and wherein the adjusting step comprises tuning a flux bias through the superconducting loop.
 18. The method of claim 1, wherein the second qubit comprises a superconducting loop and wherein the initializing step comprises tuning a flux through the superconducting loop such that a state of the second qubit tunnels to the initial classical state.
 19. The method of claim 1, wherein the coupling step comprises: increasing a coupling strength of a coupling between the first qubit and the second qubit from a minimum value to a predetermined value; and reducing the coupling strength from the predetermined value to about the minimum value.
 20. The method of claim 19, wherein the minimum value of the coupling strength is about zero and the predetermined value of the coupling strength is a maximum coupling strength of the coupling.
 21. The method of claim 19, wherein the predetermined value of the coupling strength is greater than one-half of a maximum coupling strength of the coupling.
 22. The method of claim 19, further comprising maintaining the coupling strength at the predetermined value for the duration t.
 23. The method of claim 1, wherein the coupling step comprises: turning on a coupling between the first qubit and the second qubit for the duration t; and turning off the coupling.
 24. The method of claim 1, wherein the coupling step is performed using an rf-SQUID.
 25. The method of claim 1, wherein the first qubit and the second qubit each comprise an rf-SQUID.
 26. The method of claim 1, wherein the first qubit and the second qubit each comprise a split junction.
 27. A method of copying a classical state of a first qubit to a second qubit, wherein the first qubit is characterized by a potential energy configuration that comprises a first tunneling barrier, and the second qubit is characterized by a potential energy configuration that comprises a second tunneling barrier; the method comprising: lowering the second tunneling barrier; coupling the first qubit and the second qubit for a duration t; and raising the second tunneling barrier.
 28. The method of claim 27, wherein a flux bias of at least one of the first qubit and the second qubit are maintained at about half a flux quantum during at least one of the lowering, coupling and raising steps.
 29. The method of claim 28, wherein the flux bias has a tolerance about equal to an amount of flux coupled from the first qubit to the second qubit.
 30. The method of claim 27, wherein a flux bias of the first qubit is within a hysteresis region of the first qubit during at least one of the lowering, coupling and raising steps.
 31. The method of claim 27, wherein at least one of the lowering and raising steps are performed adiabatically.
 32. The method of claim 27, wherein the coupling step comprises ferromagnetically coupling the first qubit and the second qubit.
 33. The method of claim 27, wherein the coupling step comprises anti-ferromagnetically coupling the first qubit and the second qubit.
 34. The method of claim 27, further comprising raising at least one of the first tunneling barrier and the second tunneling barrier prior to the lowering step.
 35. The method of claim 27, further comprising reducing a coupling strength between the first qubit and a device coupled to the first qubit to about zero prior to the lowering step.
 36. The method of claim 27, further comprising reducing a coupling strength between the second qubit and a device coupled to the second qubit to about zero prior to the lowering step.
 37. The method of claim 27, wherein the potential energy configuration of the first qubit is a bistable potential and the potential energy configuration of the second qubit is a bistable potential.
 38. The method of claim 37, wherein the bistable potential of at least one of the first qubit and the second qubit comprises two nearly degenerate energy levels.
 39. The method of claim 27, wherein the raising step comprises applying a transverse magnetic field to at least one of the first qubit and the second qubit.
 40. The method of claim 27, wherein the second qubit has a classical state and wherein the raising step comprises preventing the classical state of the second qubit from tunneling.
 41. The method of claim 27, wherein a time duration t₂ of the lowering step is greater than about 1 μs.
 42. The method of claim 27, wherein a time duration t₂ of the lowering step is between about 100 ns and 1 ms.
 43. The method of claim 27, wherein the second qubit does not tunnel during at least one of the lowering step and the raising step.
 44. The method of claim 27, wherein a time duration t₃ of the raising step is greater than about 1 μs.
 45. The method of claim 27, wherein a time duration t₃ of the raising step is between about 100 ns and 1 ms.
 46. The method of claim 27, wherein the second qubit comprises a split junction and wherein the lowering step and the raising step each comprise tuning a flux in the split junction.
 47. The method of claim 27, wherein the coupling step comprises: increasing a coupling strength of a coupling between the first qubit and the second qubit from a minimum coupling strength to a predetermined coupling strength; maintaining the coupling strength at the predetermined coupling strength for the duration t; and decreasing the coupling strength from the predetermined coupling strength to the minimum coupling strength.
 48. The method of claim 47, wherein the minimum coupling strength is about zero and the predetermined coupling strength is a maximum coupling strength of the coupling.
 49. The method of claim 47, wherein the predetermined coupling strength is greater than one-half of a maximum coupling strength of the coupling.
 50. The method of claim 47, further comprising maintaining the coupling strength at the predetermined coupling strength for the duration t.
 51. The method of claim 27, wherein the coupling step comprises: turning on a coupling between the first qubit and the second qubit for the duration t; and turning off the coupling.
 52. The method of claim 27, wherein the coupling step is performed using an rf-SQUID.
 53. The method of claim 27, wherein the duration t is between about 1 μs and 100 μs.
 54. The method of claim 27, wherein the duration t is greater than about 1 ns.
 55. The method of claim 27, wherein the first qubit and the second qubit each comprise an rf-SQUID.
 56. The method of claim 27, wherein the coupling step comprises tunneling through the second tunneling barrier such that a final classical state of the second qubit matches the classical state of the first qubit.
 57. A method for reading out a classical state of a qubit in an array of qubits, the array comprising a plurality of perimeter qubits and a plurality of interior qubits, the method comprising: initializing a classical state of a perimeter qubit, wherein the perimeter qubit has an associated readout device; copying a classical state of an interior qubit to the perimeter qubit; reading out the classical state of the interior qubit by reading out the classical state of the perimeter qubit, wherein the perimeter qubit is coupled to the interior qubit via a coupling device having a coupling strength that is adjustable between a minimum coupling strength and a predetermined coupling strength.
 58. The method of claim 57, wherein the minimum coupling strength is about zero and the predetermined coupling strength is a maximum coupling strength of the coupling.
 59. The method of claim 57, wherein the predetermined coupling strength is greater than one-half of a maximum coupling strength of the coupling.
 60. The method of claim 57, further comprising repeating the initializing, copying and interior qubit readout steps for each of a plurality of interior qubits.
 61. The method of claim 57, wherein the initializing step comprises initializing the perimeter qubit to an arbitrary classical state.
 62. The method of claim 57, wherein the perimeter qubit comprises a superconducting loop and wherein the initializing step comprises tuning a flux through the superconducting loop such that a state of the perimeter qubit tunnels to an arbitrary classical state.
 63. The method of claim 57, wherein the coupling device ferromagnetically couples the interior qubit to the perimeter qubit and wherein the copying step comprises copying the classical state of the interior qubit to the perimeter qubit via the ferromagnetic coupling.
 64. The method of claim 57, wherein the coupling device anti-ferromagnetically couples the interior qubit to the perimeter qubit and wherein the copying step comprises copying the classical state of the interior qubit to the perimeter qubit via the anti-ferromagnetic coupling.
 65. The method of claim 57, wherein the perimeter qubit is characterized by a potential energy configuration comprising a tunneling barrier, and wherein the copying step comprises: lowering the tunneling barrier; coupling the interior qubit and the perimeter qubit for a duration t; and raising the tunneling barrier.
 66. The method of claim 65, wherein at least one of the lowering and the raising steps is performed adiabatically.
 67. The method of claim 65, further comprising maintaining a flux bias of at least one of the interior qubit and the perimeter qubit at about half a flux quantum during the copying step.
 68. The method of claim 57, wherein the perimeter qubit is not adjacent to the interior qubit and the copying step comprises copying the classical state of the interior qubit to at least one intermediate qubit located between the interior qubit and the perimeter qubit.
 69. The method of claim 57, wherein the initializing, copying and interior qubit readout steps are performed more than once.
 70. The method of claim 57, wherein the array is a two-dimensional array of flux qubits.
 71. The method of claim 57, wherein the coupling device is configured to controllably ferromagnetically or anti-ferromagnetically couple the interior qubit to the perimeter qubit.
 72. The method of claim 57, wherein at least one of the interior qubit, the perimeter qubit and the coupling device comprises an rf-SQUID.
 73. The method of claim 57, wherein at least one of the interior qubit and the perimeter qubit comprises a split junction.
 74. The method of claim 57, wherein: the initializing step comprises initializing the perimeter qubit to an initial classical state; the perimeter qubit is characterized by a potential energy configuration comprising a first potential well having a first potential minimum and a second potential well having a second potential minimum; the initial classical state is located in the first potential well; and the copying step comprises: adjusting the first potential minimum to a third potential minimum that is higher than the second potential minimum; and coupling the interior qubit and the perimeter qubit for a duration t.
 75. A method of copying a classical state of a first qubit means to a second qubit means, the method comprising: means for coupling the first qubit means to the second qubit means; means for adjusting at least one of a tunneling barrier of the first qubit means and a tunneling barrier of the second qubit means; and means for adjusting a symmetry of a potential energy configuration of at least one of the first qubit means and the second qubit means.
 76. The method of claim 75, wherein the means for adjusting the tunneling barrier comprises means for adjusting a Josephson energy of at least one of the first qubit means and the second qubit means.
 77. The method of claim 75, wherein the means for adjusting the tunneling barrier comprises means for tuning a magnetic field bias transverse to at least one of the first qubit means and the second qubit means.
 78. The method of claim 75, wherein the means for adjusting the symmetry of the potential energy configuration comprises means for tuning a flux bias of the first qubit means and the second qubit means.
 79. The method of claim 75, wherein the means for coupling comprises means for tuning a coupling strength of the coupling means.
 80. A system for copying a classical state of a first qubit to a second qubit, wherein the first qubit is characterized by a potential energy configuration that comprises a first tunneling barrier, and the second qubit is characterized by a potential energy configuration that comprises a second tunneling barrier; the system comprising: a first barrier adjustment module, comprising instructions for lowering the second tunneling barrier; a coupling module, comprising instructions for coupling the first qubit to the second qubit; and a second barrier adjustment module, comprising instructions for raising the second tunneling barrier.
 81. The system of claim 57, wherein the first barrier adjustment module and the second barrier adjustment module comprise a single module.
 82. The system of claim 57, further comprising a flux module, comprising instructions for maintaining a flux bias of at least one of the first qubit and the second qubit at about half a flux quantum during at least a portion of the copying process.
 83. A computer-readable medium storing executable instructions for: initializing a first qubit to an initial classical state, wherein the first qubit has a potential energy configuration comprising a first potential well having a first potential minimum and a second potential well having a second potential minimum, and the initial classical state is located in the first potential well; adjusting the first potential minimum of the first potential well to a third potential minimum that is higher than the second potential minimum of the second potential well; and coupling the first qubit and a second qubit for a duration t. 